Английская Википедия:E8 lattice

In mathematics, the EШаблон:Sub lattice is a special lattice in RШаблон:Sup. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the [[E8 (mathematics)|EШаблон:Sub root system]].

The norm^[1] of the EШаблон:Sub lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8. The existence of such a form was first shown by H. J. S. Smith in 1867,^[2] and the first explicit construction of this quadratic form was given by Korkin and Zolotarev in 1873.^[3] The EШаблон:Sub lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900.^[4]

Lattice points

The EШаблон:Sub lattice is a discrete subgroup of RШаблон:Sup of full rank (i.e. it spans all of RШаблон:Sup). It can be given explicitly by the set of points ΓШаблон:Sub ⊂ RШаблон:Sup such that

all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and
the sum of the eight coordinates is an even integer.

In symbols,

<math>\Gamma_8 = \left\{(x_i) \in \mathbb Z^8 \cup (\mathbb Z + \tfrac{1}{2})^8 : {\textstyle\sum_i} x_i \equiv 0\;(\mbox{mod }2)\right\}.</math>

It is not hard to check that the sum of two lattice points is another lattice point, so that ΓШаблон:Sub is indeed a subgroup.

An alternative description of the EШаблон:Sub lattice which is sometimes convenient is the set of all points in Γ′Шаблон:Sub ⊂ RШаблон:Sup such that

all the coordinates are integers and the sum of the coordinates is even, or
all the coordinates are half-integers and the sum of the coordinates is odd.

In symbols,

<math>\Gamma_8' = \left\{(x_i) \in \mathbb Z^8 \cup (\mathbb Z + \tfrac{1}{2})^8 : {{\textstyle\sum_i} x_i} \equiv 2x_1 \equiv 2x_2 \equiv 2x_3 \equiv 2x_4 \equiv 2x_5 \equiv 2x_6 \equiv 2x_7 \equiv 2x_8\;(\mbox{mod }2)\right\}.</math>

<math>\Gamma_8' = \left\{(x_i) \in \mathbb Z^8 : {{\textstyle\sum_i} x_i} \equiv 0(\mbox{mod }2)\right\}

\cup \left\{(x_i) \in (\mathbb Z + \tfrac{1}{2})^8 : {{\textstyle\sum_i} x_i} \equiv 1(\mbox{mod }2)\right\}.</math>

The lattices ΓШаблон:Sub and Γ′Шаблон:Sub are isomorphic and one may pass from one to the other by changing the signs of any odd number of half-integer coordinates. The lattice ΓШаблон:Sub is sometimes called the even coordinate system for EШаблон:Sub while the lattice Γ′Шаблон:Sub is called the odd coordinate system. Unless we specify otherwise we shall work in the even coordinate system.

Properties

The EШаблон:Sub lattice ΓШаблон:Sub can be characterized as the unique lattice in RШаблон:Sup with the following properties:

It is integral, meaning that all scalar products of lattice elements are integers.
It is unimodular, meaning that it is integral, and can be generated by the columns of an 8×8 matrix with determinant ±1 (i.e. the volume of the fundamental parallelotope of the lattice is 1). Equivalently, ΓШаблон:Sub is self-dual, meaning it is equal to its dual lattice.
It is even, meaning that the norm^[1] of any lattice vector is even.

Even unimodular lattices can occur only in dimensions divisible by 8. In dimension 16 there are two such lattices: ΓШаблон:Sub ⊕ ΓШаблон:Sub and ΓШаблон:Sub (constructed in an analogous fashion to ΓШаблон:Sub. In dimension 24 there are 24 such lattices, called Niemeier lattices. The most important of these is the Leech lattice.

One possible basis for ΓШаблон:Sub is given by the columns of the (upper triangular) matrix

<math>\left[\begin{matrix}

2 & -1 & 0 & 0 & 0 & 0 & 0 & 1/2 \\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 1/2 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 0 & 1 & -1 & 0 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 \end{matrix}\right]</math> ΓШаблон:Sub is then the integral span of these vectors. All other possible bases are obtained from this one by right multiplication by elements of GL(8,Z).

The shortest nonzero vectors in ΓШаблон:Sub have length equal to √2. There are 240 such vectors:

All half-integer (can only be ±1/2):
- All positive or all negative: 2
- Four positive, four negative: (8*7*6*5)/(4*3*2*1)=70
- Two of one, six of the other: 2*(8*7)/(2*1) = 56
All integer (can only be 0, ±1):
- Two ±1, six zeroes: 4*(8*7)/(2*1)=112

These form a root system of type [[E8 (mathematics)|EШаблон:Sub]]. The lattice ΓШаблон:Sub is equal to the EШаблон:Sub root lattice, meaning that it is given by the integral span of the 240 roots. Any choice of 8 simple roots gives a basis for ΓШаблон:Sub.

Symmetry group

The automorphism group (or symmetry group) of a lattice in RШаблон:Sup is defined as the subgroup of the orthogonal group O(n) that preserves the lattice. The symmetry group of the EШаблон:Sub lattice is the Weyl/Coxeter group of type EШаблон:Sub. This is the group generated by reflections in the hyperplanes orthogonal to the 240 roots of the lattice. Its order is given by

<math>|W(\mathrm{E}_8)| = 696729600 = 4!\cdot 6!\cdot 8!</math>

The EШаблон:Sub Weyl group contains a subgroup of order 128·8! consisting of all permutations of the coordinates and all even sign changes. This subgroup is the Weyl group of type DШаблон:Sub. The full EШаблон:Sub Weyl group is generated by this subgroup and the block diagonal matrix HШаблон:Sub⊕HШаблон:Sub where HШаблон:Sub is the Hadamard matrix

<math>H_4 = \tfrac{1}{2}\left[\begin{smallmatrix}

1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\\ \end{smallmatrix}\right].</math>

Geometry

See [[5 21 honeycomb|5Шаблон:Sub honeycomb]]

The EШаблон:Sub lattice points are the vertices of the [[5 21 honeycomb|5Шаблон:Sub]] honeycomb, which is composed of regular 8-simplex and 8-orthoplex facets. This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure^[4] (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes). In Coxeter's notation,^[5] Gosset's honeycomb is denoted by 5Шаблон:Sub and has the Coxeter-Dynkin diagram:

Шаблон:CDD

This honeycomb is highly regular in the sense that its symmetry group (the affine <math>{\tilde{E}}_8</math> Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.

The vertex figure of Gosset's honeycomb is the semiregular [[E8 polytope|EШаблон:Sub polytope]] (4Шаблон:Sub in Coxeter's notation) given by the convex hull of the 240 roots of the EШаблон:Sub lattice.

Each point of the EШаблон:Sub lattice is surrounded by 2160 8-orthoplexes and 17280 8-simplices. The 2160 deep holes near the origin are exactly the halves of the norm 4 lattice points. The 17520 norm 8 lattice points fall into two classes (two orbits under the action of the EШаблон:Sub automorphism group): 240 are twice the norm 2 lattice points while 17280 are 3 times the shallow holes surrounding the origin.

A hole in a lattice is a point in the ambient Euclidean space whose distance to the nearest lattice point is a local maximum. (In a lattice defined as a uniform honeycomb these points correspond to the centers of the facets volumes.) A deep hole is one whose distance to the lattice is a global maximum. There are two types of holes in the EШаблон:Sub lattice:

Deep holes such as the point (1,0,0,0,0,0,0,0) are at a distance of 1 from the nearest lattice points. There are 16 lattice points at this distance which form the vertices of an 8-orthoplex centered at the hole (the Delaunay cell of the hole).
Shallow holes such as the point <math>(\tfrac{5}{6}, \tfrac{1}{6}, \tfrac{1}{6}, \tfrac{1}{6}, \tfrac{1}{6}, \tfrac{1}{6}, \tfrac{1}{6}, \tfrac{1}{6})</math> are at a distance of <math>\tfrac{2\sqrt 2}{3}</math> from the nearest lattice points. There are 9 lattice points at this distance forming the vertices of an 8-simplex centered at the hole.

Sphere packings and kissing numbers

The EШаблон:Sub lattice is remarkable in that it gives optimal solutions to the sphere packing problem and the kissing number problem in 8 dimensions.

The sphere packing problem asks what is the densest way to pack (solid) n-dimensional spheres of a fixed radius in RШаблон:Sup so that no two spheres overlap. Lattice packings are special types of sphere packings where the spheres are centered at the points of a lattice. Placing spheres of radius 1/Шаблон:Radic at the points of the EШаблон:Sub lattice gives a lattice packing in RШаблон:Sup with a density of

A 1935 paper of Hans Frederick Blichfeldt proved that this is the maximum density that can be achieved by a lattice packing in 8 dimensions.^[6] Furthermore, the EШаблон:Sub lattice is the unique lattice (up to isometries and rescalings) with this density.^[7] Maryna Viazovska proved in 2016 that this density is, in fact, optimal even among irregular packings.^[8]^[9]

The kissing number problem asks what is the maximum number of spheres of a fixed radius that can touch (or "kiss") a central sphere of the same radius. In the EШаблон:Sub lattice packing mentioned above any given sphere touches 240 neighboring spheres. This is because there are 240 lattice vectors of minimum nonzero norm (the roots of the EШаблон:Sub lattice). It was shown in 1979 that this is the maximum possible number in 8 dimensions.^[10]^[11]

The sphere packing problem and the kissing number problem are remarkably difficult and optimal solutions are only known in 1, 2, 3, 8, and 24 dimensions (plus dimension 4 for the kissing number problem). The fact that solutions are known in dimensions 8 and 24 follows in part from the special properties of the EШаблон:Sub lattice and its 24-dimensional cousin, the Leech lattice.

Theta function

One can associate to any (positive-definite) lattice Λ a theta function given by

<math>\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0.</math>

The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2. The theta function of an integral lattice is often written as a power series in <math>q = e^{i\pi\tau}</math> so that the coefficient of qШаблон:Sup gives the number of lattice vectors of norm n.

Up to normalization, there is a unique modular form of weight 4 and level 1: the Eisenstein series GШаблон:Sub(τ). The theta function for the EШаблон:Sub lattice must then be proportional to GШаблон:Sub(τ). The normalization can be fixed by noting that there is a unique vector of norm 0. This gives

<math>\Theta_{\Gamma_8}(\tau) = 1 + 240\sum_{n=1}^\infty \sigma_3(n) q^{2n}</math>

where σШаблон:Sub(n) is the divisor function. It follows that the number of EШаблон:Sub lattice vectors of norm 2n is 240 times the sum of the cubes of the divisors of n. The first few terms of this series are given by Шаблон:OEIS:

<math>\Theta_{\Gamma_8}(\tau) = 1 + 240\,q^2 + 2160\,q^4 + 6720\,q^6 + 17520\,q^8 + 30240\, q^{10} + 60480\,q^{12} + O(q^{14}).</math>

The EШаблон:Sub theta function may be written in terms of the Jacobi theta functions as follows:

<math>\Theta_{\Gamma_8}(\tau) = \frac{1}{2}\left(\theta_2(q)^8 + \theta_3(q)^8 + \theta_4(q)^8\right)</math>

where

<math>

\theta_2(q) = \sum_{n=-\infty}^{\infty}q^{(n+\frac{1}{2})^2}\qquad \theta_3(q) = \sum_{n=-\infty}^{\infty}q^{n^2}\qquad \theta_4(q) = \sum_{n=-\infty}^{\infty}(-1)^n q^{n^2}. </math> Note that the j-function can be expressed as,

<math>j(\tau) \,=\, 32\,\frac{\left(\theta_2(q)^8+\theta_3(q)^8+\theta_4(q)^8\right)^3}{\left(\theta_2(q)\,\theta_3(q)\,\theta_4(q)\right)^8}</math>

Other constructions

Hamming code

The EШаблон:Sub lattice is very closely related to the (extended) Hamming code H(8,4) and can, in fact, be constructed from it. The Hamming code H(8,4) is a binary code of length 8 and rank 4; that is, it is a 4-dimensional subspace of the finite vector space (FШаблон:Sub)Шаблон:Sup. Writing elements of (FШаблон:Sub)Шаблон:Sup as 8-bit integers in hexadecimal, the code H(8,4) can by given explicitly as the set

{00, 0F, 33, 3C, 55, 5A, 66, 69, 96, 99, A5, AA, C3, CC, F0, FF}.

The code H(8,4) is significant partly because it is a Type II self-dual code. It has a minimum Hamming weight 4, meaning that any two codewords differ by at least 4 bits. It is the largest length 8 binary code with this property.

One can construct a lattice Λ from a binary code C of length n by taking the set of all vectors x in ZШаблон:Sup such that x is congruent (modulo 2) to a codeword of C.^[12] It is often convenient to rescale Λ by a factor of 1/Шаблон:Radic,

<math>\Lambda = \tfrac{1}{\sqrt 2}\left\{x \in \mathbb Z^n : x\,\bmod\,2 \in C\right\}.</math>

Applying this construction a Type II self-dual code gives an even, unimodular lattice. In particular, applying it to the Hamming code H(8,4) gives an EШаблон:Sub lattice. It is not entirely trivial, however, to find an explicit isomorphism between this lattice and the lattice ΓШаблон:Sub defined above.

Integral octonions

The EШаблон:Sub lattice is also closely related to the nonassociative algebra of real octonions O. It is possible to define the concept of an integral octonion analogous to that of an integral quaternion. The integral octonions naturally form a lattice inside O. This lattice is just a rescaled EШаблон:Sub lattice. (The minimum norm in the integral octonion lattice is 1 rather than 2). Embedded in the octonions in this manner the EШаблон:Sub lattice takes on the structure of a nonassociative ring.

Fixing a basis (1, i, j, k, ℓ, ℓi, ℓj, ℓk) of unit octonions, one can define the integral octonions as a maximal order containing this basis. (One must, of course, extend the definitions of order and ring to include the nonassociative case). This amounts to finding the largest subring of O containing the units on which the expressions x*x (the norm of x) and x + x* (twice the real part of x) are integer-valued. There are actually seven such maximal orders, one corresponding to each of the seven imaginary units. However, all seven maximal orders are isomorphic. One such maximal order is generated by the octonions i, j, and Шаблон:Sfrac (i + j + k + ℓ).

A detailed account of the integral octonions and their relation to the EШаблон:Sub lattice can be found in Conway and Smith (2003).

Example definition of integral octonions

Consider octonion multiplication defined by triads: 137, 267, 457, 125, 243, 416, 356. Then integral octonions form vectors:

1) <math>\pm e_i</math>, i=0, 1, ..., 7

2) <math>\pm e_0\pm e_a\pm e_b\pm e_c</math>, indexes abc run through the seven triads 124, 235, 346, 457, 561, 672, 713

3) <math>\pm e_p\pm e_q\pm e_r\pm e_s</math>, indexes pqrs run through the seven tetrads 3567, 1467, 1257, 1236, 2347, 1345, 2456.

Imaginary octonions in this set, namely 14 from 1) and 7*16=112 from 3), form the roots of the Lie algebra <math>E_7</math>. Along with the remaining 2+112 vectors we obtain 240 vectors that form roots of Lie algebra <math>E_8</math>.^[13]

Applications

In 1982 Michael Freedman produced an example of a topological 4-manifold, called the [[E8 manifold|EШаблон:Sub manifold]], whose intersection form is given by the EШаблон:Sub lattice. This manifold is an example of a topological manifold which admits no smooth structure and is not even triangulable.

In string theory, the heterotic string is a peculiar hybrid of a 26-dimensional bosonic string and a 10-dimensional superstring. In order for the theory to work correctly, the 16 mismatched dimensions must be compactified on an even, unimodular lattice of rank 16. There are two such lattices: ΓШаблон:Sub>⊕ΓШаблон:Sub and ΓШаблон:Sub (constructed in a fashion analogous to that of ΓШаблон:Sub). These lead to two version of the heterotic string known as the EШаблон:Sub×EШаблон:Sub heterotic string and the SO(32) heterotic string.

References

Шаблон:Reflist

Шаблон:Cite book
Шаблон:Cite book Chapter 9 contains a discussion of the integral octonions and the EШаблон:Sub lattice.

↑ ^1,0 ^1,1 In this article, the norm of a vector refers to its length squared (the square of the ordinary norm).
↑ Шаблон:Cite journal
↑ Шаблон:Cite journal
↑ ^4,0 ^4,1 Шаблон:Cite journal
↑ Шаблон:Cite book
↑ Шаблон:Cite journal
↑ Шаблон:Cite conference
↑ Шаблон:Cite news
↑ Шаблон:Cite arXiv
↑ Шаблон:Cite journal
↑ Шаблон:Cite journal This is also Chapter 13 of Conway and Sloane (1998).
↑ This is the so-called "Construction A" in Conway and Sloane (1998). See §2 of Ch. 5.
↑ Шаблон:Cite arXiv

[norm-1] 1,0 ^1,1 In this article, the norm of a vector refers to its length squared (the square of the ordinary norm).

[Smith-2] Шаблон:Cite journal

[3] Шаблон:Cite journal

[gosset-4] 4,0 ^4,1 Шаблон:Cite journal

[coxeter-5] Шаблон:Cite book

[6] Шаблон:Cite journal

[7] Шаблон:Cite conference

[quanta-8] Шаблон:Cite news

[9] Шаблон:Cite arXiv

[10] Шаблон:Cite journal

[11] Шаблон:Cite journal This is also Chapter 13 of Conway and Sloane (1998).

[12] This is the so-called "Construction A" in Conway and Sloane (1998). See §2 of Ch. 5.

[13] Шаблон:Cite arXiv

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[5]

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Английская Википедия:E8 lattice

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